The angular momentum analysis
A large lawn sprinkler with four identical arms is to be converted into a turbine to generate electric power by attaching a generator to its rotating head, as shown in the below figure. Water enters the sprinkler from the base, along the axis of rotation, at a rate of 20 L/s and leaves the nozzles in the tangential direction. The sprinkler rotates at a rate of 300 rpm in a horizontal plane. The diameter of each jet is 1 cm and the normal distance between the axis of rotation and the center of each nozzle is 0.6 m. Note the relative motion between the jets and nozzles.
1. Use the angular momentum analysis to estimate the electric power produced. Explain your work details.
2. Conduct a parametric study to investigate the effects of operating variables that influence the generated power. These parameters may include the number of nozzles, angle of nozzles, water flow rate, sprinkler rotational speed (rpm), etc. Support your discussion with plots and trends. Select wide and suitable ranges of variables.
3. Investigate the optimum operating conditions (best combination) for maximum power production. Suggest design modifications, if possible.
Sample Solution
The electric power produced by a turbine can be estimated using angular momentum analysis. This involves calculating the change in rotational kinetic energy of the turbine as it rotates and converts this into useful electrical power. The equation used to calculate the angular momentum is: $P= \frac{1}{2}I\omega^2$ , where P is the power output, I is the moment of inertia, and $\omega$ is the angular velocity (Kivanc 2019).
In order to use this equation, we must first calculate the moment of inertia for our system. Since all four arms are identical, we can simplify our calculation by considering just one arm. The moment of inertia for a single arm can then be calculated as: $I = \frac mr^2$, where m is the mass of an arm and r its radius (Kivanc 2019). Considering that each nozzle has 1 cm diameter, then each arm will have 0.6m radius, so if water with density 1000 kg/m3 enters from base at 20L/s, then each arm will have a mass 2kg per second.
Now that we have determined all parameters needed to calculate Moment of Inertia (Mass = 2kg per sec; Radius = 0.6m), we can now substitute these values into equation for Moment of Inertia: $I=(2 kg / s)(0.6 m)^2=0.72 (kg m^2 / s)$.
Next step would be to find out Angular Velocity ($\omega$ ) which in this case will equal 300 rpm or 50 radians per second $(50rad/s)$. Substituting these two values into Power Equation : $P= \frac{1}{2}(0.72kg/s)(50rad/s) ^ 2=180 W.$
Thus ,Angular Momentum Analysis gives us an estimate that converting this particular lawn sprinkler by attaching a generator at its rotating head should generate approximately 180W electrical power .
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